Astronomy
&
Astrophysics
A&A 367, 582–587 (2001)
DOI: 10.1051/0004-6361:20000428
c ESO 2001
On the minimum and maximum mass of neutron stars
and the delayed collapse
K. Strobel and M. K. Weigel
Sektion Physik, Ludwig-Maximilians Universität München, Am Coulombwall 1, 85748 Garching, Germany
Received 23 September 2000 / Accepted 28 November 2000
Abstract. The minimum and maximum mass of protoneutron stars and neutron stars are investigated. The hot
dense matter is described by relativistic (including hyperons) and non-relativistic equations of state. We show
that the minimum mass (∼ 0.88–1.28 M ) of a neutron star is determined by the earliest stage of its evolution
and is nearly unaffected by the presence of hyperons. The maximum mass of a neutron star is limited by the
protoneutron star or hot neutron star stage. Further we find that the delayed collapse of a neutron star into a
black hole during deleptonization is not only possible for equations of state with softening components, as for
instance, hyperons, meson condensates etc., but also for neutron stars with a pure nucleonic-leptonic equation of
state.
Key words. stars: evolution – stars: neutron – dense matter – equation of state
1. Introduction
It is expected that a neutron star (NS) is born as a result
of the gravitational collapse of the iron core of a massive
(M > 8 M ) evolved star (e.g. Bethe 1990). Shortly after
core bounce (some 10 ms) a hot, lepton rich NS, called
protoneutron star (PNS), is formed. This PNS evolves in
some minutes into a cold, lepton pure NS due to the loss
of neutrinos (e.g. Burrows & Lattimer 1986; Keil & Janka
1995; Pons et al. 1999). During these first minutes the PNS
can collapse delayed to a black hole (BH) if its initial mass
is high enough, either due to the loss of neutrinos or due
to post bounce accretion.
The aim of this work is to study the minimum and the
maximum mass of NSs and the possibility of BH formation during the deleptonization period. Recently Gondek
et al. (1997, 1998), Goussard et al. (1998) and Strobel
et al. (1999a) have calculated limits on the minimum mass
of NSs, using equations of state (EOSs) including nucleons (n, p) and leptons. In this paper refinements to these
approaches are given by calculating PNSs for a large sample of EOSs using a non-relativistic Thomas-Fermi model
(Myers & Świa̧tecki 1996; Strobel et al. 1999b,a) and a
relativistic Hartree model (Serot & Walecka 1986; Schäfer
1997) including hyperons. Limits on the maximum mass
of NSs were studied by numerous authors (e.g. Takatsuka
1995; Bombaci 1996; Ellis et al. 1996; Prakash et al.
1997; Gondek et al. 1998) using different kinds of EOSs.
Send offprint requests to: K. Strobel,
e-mail: [email protected]
All these investigations start with models of PNSs about
1–3 s after core bounce (see Sect. 2 for more details). In
this paper the earliest stage of a PNS is taken additionally into account, which is reached about 100 ms after
core bounce, to calculate the maximum mass of a NS (see
Sect. 2 for more details).
2. Inside a protoneutron star
As pointed out in the Introduction a special topic of this
investigation is the more detailed incorporation of the earliest stage of the PNS, about 0.1 – 1 s after core bounce
(Burrows & Lattimer 1986; Burrows et al. 1995; Keil et al.
1996; Pons et al. 1999). This early type PNS is characterized by a hot shocked envelope with an entropy per
baryon of s ∼ 4 − 6 (in units of the Boltzmann constant,
kB ) for baryon number densities n < nenv , an unshocked
core with s ∼ 1 – 2 for densities n > ncore , and a transition region between these densities (Burrows et al. 1995;
Keil et al. 1996). For the description of this early stage
of the PNS we use, in accordance with the investigations
of Burrows & Lattimer (1988), Burrows et al. (1995) and
Keil et al. (1996), baryon densities in the range of 0.002–
0.02 fm−3 for nenv and 0.1 fm−3 for ncore . The entropy
per baryon is chosen between 5–6 in the envelope and between 1–2 in the core, respectively. The investigation is
performed for the early type PNS models with trapped
neutrinos using a constant lepton number (Yl = 0.4) for
densities above n = 6 10−4 fm−3 .
We assume that post bounce accretion onto the protoneutron star stops within the first 500 ms after core
Article published by EDP Sciences and available at http://www.aanda.org or
http://dx.doi.org/10.1051/0004-6361:20000428
K. Strobel and M. K. Weigel: On the minimum and maximum mass of neutron stars and the delayed collapse
Table 1. Properties of cold symmetric nuclear matter. The entries are: energy per baryon, u; saturation density, n0 ; incompressibility, K∞ ; symmetry energy, J; effective nucleon mass,
m∗ /m
TF
GM1
GM3
NL1
u
[MeV]
n0
[fm−3 ]
K∞
[MeV]
J
[MeV]
m∗ /m
−16.24
−16.30
−16.30
−16.42
0.161
0.153
0.153
0.152
234
300
240
212
32.7
32.5
32.5
43.5
0.87
0.70
0.78
0.57
583
Table 2. Coupling constants for the relativistic Hartree models. The coupling constants are given for the following meson
masses: mσ = 550 MeV, mω = 783 MeV and mρ = 770 MeV
(the coupling constants are converted according to these meson
masses)
GM1
GM3
NL1
gσ2 /4π
gω2 /4π
gρ2 /4π
103 b̄
103 c̄
7.288
6.139
10.2099
8.959
6.041
13.6108
5.346
5.807
8.0260
2.947
8.659
2.4578
−1.070
−2.421
−3.4334
2000.0
1800.0
TF
NL1
NL1Hyp
1600.0
1400.0
−3
P [ MeV fm ]
bounce. The amount of accreted matter, and the time at
which the accretion through the shock front stops are still
open questions. The amount of accreted matter after core
bounce ranges from 0.5 M during the first 10 ms or so
to 0.001 to 0.15 M during the following 100 ms or so,
but in most calculations significant accretion stops about
500 ms after core bounce (for a further discussion of this
topic see e.g. Burrows & Lattimer 1988; Chevalier 1989;
Herant et al. 1992; Burrows et al. 1995; Janka & Müller
1996; Mezzacappa et al. 1998; Fryer & Heger 2000).
About 1–3 s after core bounce the PNS has a nearly
constant entropy per baryon (s ∼ 2) and an approximately constant lepton number (Yl ∼ 0.4) for densities n > 6 10−4 fm−3 . The next stage in the lifetime of
a NS is reached after about 10–30 s, where the hot NS
(s ∼ 2) is now transparent to neutrinos. During the following minutes this hot NS evolves into a cold NS. For a
more detailed description of the evolution of PNSs see, for
instance, Prakash et al. (1997) and Strobel et al. (1999a).
The EOSs used in this paper for the description of
PNSs are: (i) A non-relativistic Thomas-Fermi model
(TF) for finite temperatures developed by Strobel et al.
(1999a,b). (ii) A relativistic Hartree model for finite temperatures developed by Schäfer (1997). The Hartree model
includes the following hyperons and nucleonic isobars:
Λ, Σ− , Σ0 , Σ+ , Ξ− , Ξ0 , ∆− , ∆0 , ∆+ , ∆++ (the Delta resonances are not included in the NL1 model).
The main properties of cold symmetric nuclear matter
of the different models are listed in Table 1 (see Strobel
et al. 1999a; Glendenning & Moszkowski 1991; Reinhard
1989). The coupling constants for the Hartree models
(GM1, GM3, NL1) are listed in Table 2. We take the
nucleon-hyperon coupling strength equal to the nucleonnucleon coupling strength for simplicity. This restriction
will not change the results for the minimum mass of a
NS, as we will show later. The influence on the maximum mass will be strong, but for this point we refer to
Huber et al. (1998), where this question is studied for a
large spectrum of different coupling constants and different nucleon-hyperon coupling strengths for cold NSs. The
EOSs for the different models with nucleonic-leptonic matter (TF, GM1, GM3 and NL1) and nuclear matter including hyperons (GM1Hyp , GM3Hyp and NL1Hyp ) are shown
in Figs. 1 and 2. The EOSs for baryon number densities
1200.0
1000.0
800.0
600.0
400.0
200.0
0.0
0.1
0.3
0.5
0.7
0.9
−3
n [ fm ]
1.1
1.3
1.5
Fig. 1. Pressure versus baryon number density in the high
density region for different EOSs. The solid curves correspond (from the bottom to the top) to the cold EOS, the
s = 1, Yl = 0.4 EOS and the s = 2, Yl = 0.4 EOS of
the TF model. The long dashed curves correspond to the cold
EOS (lower) and the s = 1, Yl = 0.4 EOS (upper) of the NL1
model without hyperons (dot-dashed lines, hyperons included)
below n = 0.1 fm−3 are taken from the investigation of
Strobel et al. (1999a).
The NL1 coupling constants lead to the stiffest EOSs
at high densities, since the high density behaviour is dominated by the strength of the ω-coupling. However the incompressibility is the smallest. In this respect one has to
keep in mind that the incompressibility determines the
stiffness only near saturation and for higher densities the
second and third derivatives of the incompressibility are
important for the stiffness (see for instance Ramschütz
et al. 1990).
3. The minimum mass of neutron stars
The minimum and maximum mass of non-rotating NSs
for the different EOSs and different choices of the transition region between the hot shocked envelope and the
unshocked core of the PNS are given in Table 3. It turned
out that the minimum mass of a NS is determined by
the earliest stage of the lifetime of a PNS. The values for
the minimum gravitational (baryonic) mass, MG (MB ),
lie in the range of 0.878–1.284 M (0.949–1.338 M ) for
584
K. Strobel and M. K. Weigel: On the minimum and maximum mass of neutron stars and the delayed collapse
Table 3. EOSs used in this paper and resulting minimum and maximum masses. The entries are: time after core bounce, t;
entropy per baryon in the envelope, senv ; entropy per baryon in the core, score ; maximum baryon number density of the envelope
correlated with the entropy per baryon in the envelope, nenv (senv ); minimum baryon number density of the core correlated with
the entropy per baryon in the core, ncore (score ); minimum baryonic mass of the PNS, MBmin ; resulting minimum gravitational
min
(T = 0); maximum baryonic mass of the PNS or NS, MBmax ; maximum gravitational mass of the cold
mass of the cold NS, MG
max
NS determined by the PNS or hot NS, MG
(T = 0); possibility of the formation of a black hole during deleptonization, BH
Model
TF
GM1
GM1Hyp
GM3
GM3Hyp
NL1
NL1Hyp
t
[s]
senv
score
nenv (senv )
[fm−3 ]
ncore (score )
[fm−3 ]
MBmin
[M ]
min
MG
(T = 0)
[M ]
MBmax
[M ]
max
MG
(T = 0)
[M ]
BH
0.1 – 1
0.1 – 1
0.1 – 1
0.1 – 1
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
0.1 – 1
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
0.1 – 1
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
0.1 – 1
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
0.1 – 1
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
1–3
10 – 30
∞
0.1 – 1
1–3
10 – 30
∞
6.0
6.0
6.0
5.0
5.0
2.0
2.0
0.0
6.0
5.0
5.0
2.0
2.0
0.0
6.0
5.0
5.0
2.0
2.0
0.0
6.0
5.0
5.0
2.0
2.0
0.0
6.0
5.0
5.0
2.0
2.0
0.0
5.0
2.0
2.0
0.0
5.0
2.0
2.0
0.0
2.0
1.0
1.0
1.0
1.0
2.0
2.0
0.0
1.0
1.0
1.0
2.0
2.0
0.0
1.0
1.0
1.0
2.0
2.0
0.0
1.0
1.0
1.0
2.0
2.0
0.0
1.0
1.0
1.0
2.0
2.0
0.0
1.0
2.0
2.0
0.0
1.0
2.0
2.0
0.0
0.002
0.002
0.005
0.01
0.02
0.005
0.005
0.01
0.005
0.005
0.01
0.005
0.005
0.01
0.005
0.005
0.01
0.005
0.005
-
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
-
1.338
0.949
1.117
1.020
1.229
1.255
1.019
1.140
1.255
1.019
1.140
1.255
1.039
1.145
1.255
1.039
1.145
1.175
1.169
-
1.284
0.878
1.021
0.940
1.114
1.206
0.988
1.100
1.206
0.987
1.100
1.205
1.005
1.104
1.205
1.005
1.104
1.111
1.104
-
2.366
2.371
2.372
2.372
2.374
2.364
2.449
2.417
2.629
2.628
2.629
2.613
2.670
2.706
2.462
2.461
2.462
2.401
2.304
2.353
2.204
2.203
2.204
2.212
2.249
2.247
2.047
2.046
2.047
2.015
1.898
1.911
3.218
3.162
3.202
3.297
3.017
2.942
2.871
2.962
1.969
2.251
2.015
1.939
1.938
1.939
1.699
2.663
(2.689)
2.399
-
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
yes
yes
yes
yes
yes
(yes)
yes
yes
-
the different models (see Table 3). They are larger by a
factor of ten than the minimum mass of a cold NS (e.g.
Strobel et al. 1997) emerging from a calculation for the
minimum mass of the cold NS based only on the solution of the Tolman-Oppenheimer-Volkoff equation with
the corresponding EOS for cold NS matter. This shift due
to the properties of early type PNSs was also recently
found by Goussard et al. (1998) and Strobel et al. (1999a).
Gondek et al. (1997, 1998) found an even smaller value
for the minimum mass (MG ∼ 0.6 M ). The reason for
this is that they use models for PNSs about 1–3 s after core bounce, where the minimum mass is lower, see
K. Strobel and M. K. Weigel: On the minimum and maximum mass of neutron stars and the delayed collapse
1600.0
2.8
1400.0
GM1
GM1Hyp
GM3
GM3Hyp
1200.0
2.75
1000.0
MG [ Msun ]
−3
P [ MeV fm ]
585
800.0
600.0
400.0
2.7
t = 0.1−1s
t = 1−3 s
t = 10−30 s
t = minutes
2.65
200.0
0.0
0.1
0.3
0.5
0.7
0.9
−3
n [ fm ]
1.1
1.3
1.5
Fig. 2. Pressure versus baryon number density in the high density region for different EOSs. The dotted curves correspond
to the cold EOS (lower) and the s = 1, Yl = 0.4 EOS (upper) of the GM1 model without hyperons (long dashed lines,
hyperons included). The dot-dashed curves correspond to the
cold EOS (lower) and the s = 1, Yl = 0.4 EOS (upper) of the
GM3 model without hyperons (short dashed lines, hyperons
included)
also Strobel et al. (1999a). Our models with hyperons included do not change the values for the minimum mass
significantly (∆ MG ≤ 0.007 M ). This result could be
expected since the maximum baryon number density inside the core of such a NS is smaller than two times the
nuclear matter density and hence the hyperon fraction is
rather small.
Let us now turn to the question, why there is a minimum mass in this early stage of the evolution. A star becomes dynamically unstable if its mean adiabatic index,
Γ̄, lies below 4/3. In the case of a cold NS this is caused
by the neutron drip at a density of n ∼ 2.6 10−4 fm−3 .
For the earliest stage of the PNS the drop below Γ̄ = 4/3
is caused by the entropy drop between the hot shocked
envelope and the unshocked core of the PNS. For a complete description of the stability criteria of neutron stars
see Shapiro & Teukolsky (1983).
4. The maximum mass of neutron stars
and the possibility of black hole formation
during deleptonization
Quite interesting is the result for the maximum gravitational mass for NSs, for which we obtain values in the
range between 1.699–2.663 M (see Table 3) for the different EOSs used in this paper, with the possibility of
the formation of a BH during deleptonization for EOSs
with a pure nucleonic-leptonic composition and EOSs
which include hyperons. For instance, for the NL1
EOS, the maximum baryonic mass for the first milliseconds, MBmax = 3.218 M , is larger than the allowed maximum baryonic mass of 3.162 M after 1–3 s, which implies
2.6
3.1
3.15
3.2
3.25
MB [ Msun ]
3.3
3.35
Fig. 3. Maximum gravitatinal versus maximum baryonic mass
for different evolutionary stages of the NL1 EOSs. The two
lines denote the mass window for the delayed collapse
the possibility of a BH formation (see Table 3 and Fig. 3).
This value also sets the boundary of 2.663 M for the
maximum gravitational mass for the cold NS. The maximum baryonic mass of the first stage is even larger than
maximum value 10–30 s after core bounce fore the NL1
EOS. This means that the delayed collapse is also possible for the totally deleptonized hot NS of the NL1 EOS. It
should be mentioned in this context that the exact entropy
profile of the hot shocked envelope of an early type PNS
plays a minor role in determining the maximum baryonic
mass, since a PNS with a constant entropy per baryon of
s = 1 throughout the whole PNS and a lepton number
of Yl = 0.4 leads to nearly the same maximum baryonic mass (e.g. for the NL1 model: MBmax = 3.218 M,
0.1–1.0 s after core bounce; MBmax = 3.219 M, for an
entropy per baryon of s = 1 throughout the whole PNS
and Yl = 0.4). This means that the maximum mass depents nearly complete on the properties of the core of the
PNS1 .
Similar behaviour is found for the other EOSs with
exception of the nucleonic-leptonic EOS GM3, where the
allowed maximum mass after 1–3 s exeeds the maximum masses of the first period. The reason why this
possibility of a BH formation was dismissed so far for a
pure nucleonic-leptonic composition (e.g. Takatsuka 1995;
Bombaci 1996; Ellis et al. 1996; Prakash et al. 1997;
Gondek et al. 1998) was that these authors start their calculations about 1–3 s after core bounce, since the following
stages allow only larger baryonic masses. One should mention in this context that for the nucleonic-leptonic EOSs
the differences in the maximum masses in the first seconds
are more than three times smaller2 (<0.056 M) than for
1
Note: Other properties of an early type PNS (e.g. the
radius) are strongly affected by the entropy profile.
2
For that reason the effect was overlooked in Strobel et al.
(1999a).
586
K. Strobel and M. K. Weigel: On the minimum and maximum mass of neutron stars and the delayed collapse
0.06
0.05
NL1
0.04
∆M [ Msun ]
0.03
0.02
0.01
GM1
TF
0
−0.01
GM3
−0.02
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
MB [ Msun ]
3
3.1 3.2 3.3
Fig. 4. Baryonic mass difference between the early type PNS
stage (0.1 – 1 s after core bounce) and the late type PNS stage
(1 – 3 s after core bounce) versus resulting maximum baryonic
mass for the different pure nucleonic-leptonic EOSs
the softer EOSs with hyperons (0.146–0.158 M ). Due
to this, for the EOSs including hyperons the possibility
of a BH formation shows up even by starting the calculation 1–3 s after core bounce (see Table 3). It turned
out, that the mass window for the delayed collapse of the
pure nucleonic-leptonic EOSs is an increasing function of
the maximum possible mass of the different models (see
Fig. 4). The EOSs including hyperons did not show a similar behaviour, the mass window is nearly constant for all
used EOSs. The minimum value of the maximum baryonic mass for the models including hyperons is reached
by the hot NS without trapped neutrinos (10–30 s after
core bounce). This, as already mentioned, can lead to a
delayed collapse of the NS during deleptonization if the
initial mass of the PNS is high enough (e.g. Baumgarte
et al. 1996; Pons et al. 1999).
It is an open question how large the maximum mass
of a NS really is. It is clear that an EOS should allow NS
masses larger than 1.4 M since measurements of pulsars
in binary systems show up values around this mass (e.g.
Thorsett & Chakrabarty 1999). If quasi periodic oscillations (e.g. van der Klis 2000) are taken into account, the
possible maximum mass of an EOS has to be larger than
1.8−2.0 M (e.g. Schaab & Weigel 1999). This whould
lead to the conclusion that only the GM1 and the NL1
EOSs (with and without hyperons) are stiff enough to allow NSs with a larger mass.
5. Discussion and conclusion
In this paper we presented a calculation of the minimum
and maximum mass of a NS taking the early evolution,
shortly after core bounce, into account. We found that
the minimum mass of a NS is limited by the earliest stage
of its evolution (∼0.1–1 s after core bounce). Therefore
the minimum gravitational mass has values between 0.878
and 1.284 M . These values lie in the same range as
those found in recent works by Goussard et al. (1998)
and Strobel et al. (1999a). Additionally we have examined the influence of hyperons on the minimum mass. As
expected it turned out that the minimum mass is nearly
unaffected by the inclusion of hyperons. Another possibility of NS production is the accretion induced collapse of a
white dwarf (Canal & Schatzman 1976), but the numerical simulation of Woosley & Baron (1992) showed that the
resulting NS will have a mass of ∼1.4 M since the white
dwarf collapses without significant mass loss after reaching the maximum mass of a white dwarf. From the results
reported it is obvious that NSs, which are born in type-II
supernovae or due to accretion induced collapse of a white
dwarf, have a minimum mass about ten times larger than
the possible minimum mass of a cold NS (∼0.1 M ).
With respect to the maximum mass of a NS calculated
by solving the Tolman-Oppenheimer-Volkoff equation for
NS matter the outcome is rather uncertain, since the EOS
in the high density region rests strongly on theoretical extrapolations. The underlying model, for instance, different many-body approximations and/or inclusion of more
massive baryons, quarks, condensates etc., can render the
results for the maximum masses significantly (e.g. Prakash
et al. 1997; Huber et al. 1998). However, as a result of our
work it turned out, that the maximum mass of a NS is
limited by the first ∼30 s of its evolution. The maximum
baryonic mass of a NS, constructed with pure nucleonicleptonic EOSs and EOSs including hyperons, is always
reached during this early epoch of the evolution.
Finally we found that BH formation during deleptonization is also possible for NSs with pure nucleonicleptonic EOSs if the initial mass is high enough. This property of NSs was expected till now to be possible only for
EOSs including softening ingredients, like hyperons, meson condensates or a quark-hadron phase transition (e.g.
Brown & Bethe 1994; Glendenning 1995). There are numerous papers on this topic (e.g. Takatsuka 1995; Bombaci
1996; Ellis et al. 1996; Prakash et al. 1997; Gondek et al.
1998), but all these investigations started their calculations about 1–3 s after core bounce. In this paper we included the earliest stage and found that the maximum
baryonic mass of the early type PNSs (∼0.1–1 s after core
bounce) could be up to 0.056 M larger than that of the
late type PNS (∼1–3 s after core bounce). Due to this
result it seems possible that a NS with a pure nucleonicleptonic EOS could also undergo delayed collapse into a
BH during the first seconds of its evolution. This question
could only be finally decided in time evolution calculations
like this including softening ingredients (e.g. Baumgarte
et al. 1996), since we have used approximate values for
the entropy per baryon and the lepton number at fixed
times. Post bounce accretion (e.g. Chevalier 1989; Herant
et al. 1992) could lead also to a delayed collapse, but it
is uncertain till now how much matter falls through the
shock front within the first second after core bounce, so
that if a cut off of a neutrino signal after a few seconds
K. Strobel and M. K. Weigel: On the minimum and maximum mass of neutron stars and the delayed collapse
is detected in the future, the two ways of BH formation
could possibly be indistinguishable.
Acknowledgements. We want to thank A. Schäfer for providing
us with his program. We want to thank H.-Th. Janka and Ch.
Schaab for many helpful discussions. One of us, K. S., gratefully
acknowledges the Bavarian State for financial support.
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